Computational complexity of $k$-stable matchings

TL;DR

This paper explores the computational complexity of finding and verifying $k$-stable matchings across different market models, revealing polynomial-time verification but model-dependent complexity for existence.

Contribution

It introduces the concept of $k$-stability in matching markets and analyzes the complexity of existence and verification problems across three models.

Findings

Verification of $k$-stability is polynomial-time in all models.

Existence of $k$-stable matchings varies in complexity depending on the model and the ratio $k/n$.

Complexity results depend on the relationship between $k$ and $n$.

Abstract

We study deviations by a group of agents in the three main types of matching markets: the house allocation, the marriage, and the roommates models. For a given instance, we call a matching $k$-stable if no other matching exists that is more beneficial to at least $k$ out of the $n$ agents. The concept generalizes the recently studied majority stability. We prove that whereas the verification of $k$-stability for a given matching is polynomial-time solvable in all three models, the complexity of deciding whether a $k$-stable matching exists depends on $\frac{k}{n}$ and is characteristic to each model.